Make a Cone from a Sector Calculator

Diagram showing how a cone is made from a circular sector - flat pattern development
Figure: A cone formed by joining the radii of a sector. The sector's radius s becomes the slant height of the cone.

To make a cone, start with a sector of central angle \(\theta\) and radius \(s\). Join points A and B, allowing point O to move upward until OA and OB coincide. The sector radius \(s\) equals the slant height of the finished cone.

Cone Dimensions to Sector

Enter base radius and height to get the flat pattern layout dimensions

Cone Geometry
-- degrees
-- cm

Results rounded to 2 decimal places. All inputs must be positive.

Mathematical Formulas

For a cone with base radius \(r\) and height \(h\):

1. Slant height (sector radius):

\[ s = \sqrt{r^2 + h^2} \]

2. Sector central angle (degrees):

\[ \theta = 360^\circ \times \frac{r}{s} = \frac{360^\circ \cdot r}{\sqrt{r^2 + h^2}} \]

Derivation: The arc length of the sector (\(\theta s\)) equals the base circumference (\(2\pi r\)). Solving for \(\theta\) in radians and converting to degrees gives the formula above.

Worked Example

Example: r = 6 cm, h = 8 cm

Step 1 – Slant height:
\(s = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\,\text{cm}\)

Step 2 – Sector angle:
\(\theta = 360 \times \dfrac{6}{10} = 360 \times 0.6 = 216^{\circ}\)

Result: Cut a sector with radius \(10\,\text{cm}\) and central angle \(216^{\circ}\). When joined, this forms a cone of base radius \(6\,\text{cm}\) and height \(8\,\text{cm}\).

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